Can I do deep learning with the 1060 or the 1070 ti?
Regarding specific choices I can't recommend, but if you are completely new, you should probably learn/code some more until you get a GPU. There is a lot to learn in machine learning before GPU speedups make a significant difference, and until then doing the computations on any old CPU would be just fine, especially if you are just starting since you won't be doing anything too complex. You will know when computational resources are your main bottleneck, and until then it shouldn't really matter too much. 

Or, you could also rent computing power from say, AWS or Google